[/quote]
,
such that 
divides![$\sum_{0 \leq i < j \leq 2024} \left[\binom{2024}{i}\binom{2034}{j} - \binom{2024}{j}\binom{2034}{i}\right]^2$](http://latex.artofproblemsolving.com/4/c/4/4c42b9f80506262b06ff41a89c911784d28e9de2.png)
,
.
with 
,
to the sides 
are 
,
is the orthocenter. 
is the midpoint of segment 
.
and 
intersect at 
.
from 
and 
intersect at 
.
are colinear
and 
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
prime and 
satisfying the equation: 
be positive real numbers with 
.
and 
satisfy 
,
is an even number.
2. 
3. 
Bracket 2
2. 
3. 
Bracket 3![$$\int_0^{1}\frac{dx}{1-\sqrt[3]{1-\sqrt{x}}}$$](http://latex.artofproblemsolving.com/d/5/e/d5e8db00985944b2eca341e37a94e34661cb5c86.png)
2. 
3. 
Bracket 4
2. 
3. 
Bracket 5
2. 
3. 
Bracket 6
2. 
3. 
Bracket 7
2. 
3. 
Bracket 8
2. 
3. 
![$$\int_0^{2\pi}\sqrt[4]{\cos^4(x)-\cos(2x)}dx$$](http://latex.artofproblemsolving.com/7/e/e/7eebb4a488fec702e0130a253ef87f0f6dfedb2d.png)
2. ![$$\int_0^1 \frac{x^5}{\sqrt[3]{1-x^3}}dx$$](http://latex.artofproblemsolving.com/6/0/5/605bb9d81d0ccff54a276c9f7bf1c1726c136398.png)
3. 
Bracket 2
2. 
3. 
Bracket 3
2. ![$$\int_0^1 \frac{dx}{(\sqrt{x}+\sqrt[3]{x})^2}$$](http://latex.artofproblemsolving.com/6/9/8/69869c0ed754c0dfacf412ac6411742ffa164a6d.png)
3. 
Bracket 4
2. 
3. 
2. 
3. 
Bracket 2
2. 
3. 
2. 
3. 
2. 
3. 
4. 
5. 
is differentiable and for all 
.
Y by FFA21
Any idea?? Diff equational system combined with Matrix theory.
Consider the equation dX/dt=X^2, where X(t) is an n×n matrix satisfying the condition detX=0. It is known that there are no solutions of this equation defined on a bounded interval, but there exist non-continuable solutions defined on unbounded intervals of the form (t ,+∞) and (−∞,t). Find n.
Consider the equation dX/dt=X^2, where X(t) is an n×n matrix satisfying the condition detX=0. It is known that there are no solutions of this equation defined on a bounded interval, but there exist non-continuable solutions defined on unbounded intervals of the form (t ,+∞) and (−∞,t). Find n.
This post has been edited 1 time. Last edited by c00lb0y, Apr 18, 2025, 7:13 AM
Reason: unsolved yet!
Reason: unsolved yet!
,
.
and true for 
.
.
,
constants: 
.
